What was already known
The known bounds were h₃(G)≤n and h₅(G)≤(n−1)/2. An n/2 diameter-four bound applied only without the triangle-free requirement.
AI involvement
Nick Kuhn posed the problem. Claude Fable 5 found the construction and proof sketch and guided formalization; GPT-5.5 implemented the Lean proof with Codex.
Fable supplied the counterexample and guided the Lean proof while Kuhn directed and published the work; the complete interaction record is insufficient for AI-autonomous.
Importance
Rules out the proposed uniform linear saving at diameter four.
Limitations
The reported Lean and comparator checks cover the formal counterexample. The public sketch is credited to Claude Fable 5; a named human full proof check or independent explanation was not located.
Technical verification
Kuhn's dated verification log reports a Lean build, axiom audit and comparator kernel check of both the repository's negation and the vendored Formal Conjectures formulation. The latter was compared with the subsequently merged statement. This check is source-reported, not reproduced here; no complete human proof check is documented.
Formal checking
Pinned build, axiom audit restricted to propext/Classical.choice/Quot.sound, statement comparator and kernel replay
- Checker
- Lean 4 v4.28.0 and the Lean comparator, run by Nick Kuhn and repository CI
- Scope
- Negation of Erdős #619 for connected finite triangle-free graphs, including the Formal Conjectures formulation
- Basis
- Reported by the linked checker
Checking a formal statement does not establish that it matches the original problem.
Human understanding
The public sketch and README explain the mechanism but credit the original sketch to Fable; an independently human-authored explanation of the full central counting argument is not established. Few is the missing-evidence default, not a finding that people cannot understand it.
Correction and publication history
Rules out the proposed uniform linear saving at diameter four.
Read this published revision